A304203 If n = Product (p_j^k_j) then a(n) = Product (p_j^prime(k_j)).

1, 4, 9, 8, 25, 36, 49, 32, 27, 100, 121, 72, 169, 196, 225, 128, 289, 108, 361, 200, 441, 484, 529, 288, 125, 676, 243, 392, 841, 900, 961, 2048, 1089, 1156, 1225, 216, 1369, 1444, 1521, 800, 1681, 1764, 1849, 968, 675, 2116, 2209, 1152, 343, 500, 2601, 1352, 2809, 972, 3025

Offset: 1

Author

Ilya Gutkovskiy, May 09 2018

Keywords

• nonn
• mult

Examples

			a(12) = a(2^2*3^1) = 2^prime(2)*3^prime(1) = 2^3*3^2 = 72.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000040, A001248, A002110, A003963, A008477, A030078, A048767, A050997, A056166 (records), A061742, A181819.

Cf. A064988 (apply prime to p), A321874 (apply prime to both p & e).

Programs

Maple

a:= n-> mul(i[1]^ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 20 2021

Mathematica

a[n_] := Times @@ (#[[1]]^Prime[#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 55}]

PARI

a(n) = my(f=factor(n)); prod(k=1, #f~, f[k,1]^prime(f[k,2])); \\ Michel Marcus, May 09 2018

PARI

apply( A304203(n)=factorback((n=factor(n))[,1],apply(prime,n[,2])), [1..50]) \\ M. F. Hasler, Nov 20 2018

Formula

a(prime(i)^k) = prime(i)^prime(k).

a(A000040(k)) = A001248(k).

a(A001248(k)) = A030078(k).

a(A030078(k)) = A050997(k).

a(A002110(k)) = A061742(k).

Multiplicative with a(p^e) = p^prime(e). - M. F. Hasler, Nov 20 2018

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^prime(k)) = 1.80728269690724154161... . - Amiram Eldar, Jan 20 2024